Question: Let $h$ be a twice differentiable function, and let $h(5)=1$, $h'(5)=0$, and $h''(5)=2$. What occurs in the graph of $h$ at the point $(5,1)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(5,1)$ is a minimum point. (Choice B) B $(5,1)$ is a maximum point. (Choice C) C There's not enough information to tell.
Explanation: Since $h'(5)=0$, we know that $x=5$ is a critical point. The second derivative test allows us to analyze what happens in the graph of $h$ at this point according to these three cases: If $h''(5)>0$, the graph of $h$ has a minimum point at $x=5$. If $h''(5)<0$, the graph of $h$ has a maximum point at $x=5$. If $h''(5)=0$, the test is inconclusive. [Why is this so?] We are given that $h''(5)=2>0$. Therefore, $(5,1)$ is a minimum point.